Updated June 08, 2015.
Q. My 5 year old daughter has a bad cough and high temperature, was sick and brought up phlegm. I took her to GP who checked her over, sounded chest and said it was clear, it was a viral cold to continue giving calpol and ibuprofen. This was 2pm. At 6pm her temperature was 103.3f (this was after her being stripped to her vest and wiped down cool air etc.) and I contacted our NHS helpline, who told me to take her to the out of hour doctors. On arrival, her temp was 40.4c. and she was stripped and given a fan to cool her. After 40mins her temp still 39.7c. The doctor sounded her and diagnosed a bad chest infection and gave her antibiotics. WHY DID MY OWN GP NOT FIND THIS 4 1/2 HOURS PREVIOUS????? Sandra, Scotland
A.

This is a common scenario that raises a lot of important questions and issues...

How do you know which doctor to believe? Don't be fooled into thinking that the second doctor was right just because they prescribed your child antibiotics.

There are three main possibilities:

Both Doctors Are Right

You could give both doctors the benefit of the doubt and believe that they were both right. That would mean that your child did indeed just have a cold virus during your first visit and then simply worsened over the next 4 hours and developed pneumonia.
It is not unreasonable that this could happen. I have seen children many times in the morning and thought they had a simple cold, only for them to develop severe ear pain later in the afternoon and have a rip roaring ear infection.

Children sometimes get worse before they get better and it can sometimes be hard to make an accurate diagnosis very early in an illness, especially on the first day that the symptoms start.

Only The Second Doctor Was Right

Another possibility, which you seem to be leaning towards, is that your doctor simply made a mistake.

It is possible that your GP missed the diagnosis of a pneumonia, especially if they didn't do a careful physical exam. Have you had problems with this doctor in the past?
Did your child have other common symptoms of pneumonia, like chest pain or trouble breathing?

Only Your Doctor Was Right

It is also very possible that your doctor was right and that the second doctor misdiagnosed your child as having a 'bad chest infection' that required antibiotics. And unless the second doctor did a chest xray to confirm the diagnosis, I wouldn't be quick to believe that your daughter really has pneumonia, especially in the middle of cold and flu season when many children have coughs and high fever. So that could mean that your child was prescribed the antibiotics unnecessarily, which is not hard to believe since over-prescribing antibiotics is such a big problem.
Have you ever gone to this after hours clinic before? Have you, or anyone you know, ever left there without a prescription for antibiotics?

In Conclusion

In situations like this, when you have two differing opinions, you shouldn't always be quick to believe one doctor just because they prescribed a medication. It is actually much easier to prescribe antibiotics then it is to consider why the child doesn't need them and convince a parent that they aren't necessary.
Remember that antibiotics don't treat viral infections and your child doesn't always need an antibiotic just because they have a high fever. It is one of the more frustrating things in medicine when you do the right thing, but patients believe another doctor who may be wrong, just because they prescribed antibiotics.

If your child continues to have fever, you might see your doctor again and request that a chest xray be done to confirm the diagnosis one way or the other.